DIPARTIMENTO DI FISICA
CORSO DI LAUREA MAGISTRALE IN FISICA
Asymptotic analysis and numerical evaluations of vertex
amplitudes in spinfoam models
TESI DI LAUREA MAGISTRALE
Candidato
Relatori
Marco Fanizza
dott. Simone Speziale
prof. Enore Guadagnini
1 A path to quantum gravity 1
1.1 Einstein-Hilbert action and perturbative quantization . . . 1
1.2 Canonical quantization of constrained systems. . . 2
1.3 Tetrad formulation . . . 3
1.4 Quantization in new variables . . . 6
1.5 Observables and transition amplitudes . . . 9
1.6 General relativity from BF theory . . . 11
1.6.1 BF Theory: symmetries and classical equations . . . 11
1.6.2 Plebanski action . . . 12
1.7 Regge calculus . . . 13
1.8 Spinfoams as path integrals for generally covariant systems . . . 14
2 Spinfoam quantization 17 2.1 Spinfoam quantization of BF theory . . . 17
2.1.1 Riemannian 3d Gravity: SU(2) BF Theory . . . 20
2.1.2 4d SU(2) BF Theory . . . 21
2.2 Coherent States. . . 22
2.2.1 Amplitudes in the Coherent Representation . . . 23
3 Coherent SU(2) 4-simplex amplitude 25 3.1 15j symbol asymptotic . . . 25
3.1.1 Geometric boundary data and 4-simplex reconstruction . . . 26
3.1.2 Characterization of the critical points . . . 28
3.1.3 Expansion around the critical points . . . 30
3.2 Numerical evaluation. . . 31
3.2.1 A trick for the computation: the factorization of the amplitude . . . 32
3.2.2 Boundary intertwiner decomposition . . . 35
3.2.3 Factorized coherent SU (2) amplitude . . . 37
3.2.4 Results and comments . . . 37
4 Unitary irreducible representations of SL(2, C) for the EPRL model 41 4.1 Unitary irreducible representation of SL(2, C) . . . 41
4.1.1 Canonical basis . . . 41
4.1.2 Linear spaces of homogenous functions . . . 42
4.1.3 Coherent states . . . 43
4.2 Clebsch-Gordan coefficients . . . 44
4.3 Asymptotic of n-invariants. . . 46
5 The Lorentzian EPRL Model 51 5.1 Linear simplicity constraint and Y map . . . 51
5.2 Partition function. . . 52
5.3 Vertex amplitude decomposition . . . 53
5.3.1 B4(jf, lf; i, k) . . . 55
5.3.2 Factorized coherent SL(2, C) Amplitude . . . 58
6 Coherent EPRL vertex amplitude 59 6.1 Asymptotic analysis . . . 59
6.1.1 Critical point equations . . . 60
6.1.2 Classification of the solutions . . . 62
6.1.3 Asymptotic for Euclidean boundary data . . . 62
6.1.4 Asymptotic for Lorentzian boundary data . . . 63
6.2 Joint criticality and selection rules . . . 66
6.3 Numerical evaluations . . . 67
6.3.1 Computational difficulties . . . 68
6.3.2 Convergence of the amplitude . . . 68
6.3.3 Results and comments . . . 70
6.4 Perspectives for quantum gravity predictions . . . 72
7 Conclusions 73 A Appendix - Graphical Calculus 75 A.1 Glossary . . . 75
A.1.1 SU(2) Symbols . . . 75
A.1.2 B3 and B4 function . . . 77
A.1.3 Coherent States. . . 77
A.1.4 Boxes and Integrations. . . 78
B Appendix - Conventions and Definitions 79 B.1 SU(2) Clebsch-Gordan Coefficients . . . 79
B.2 Boost matrix elements for general unitary irreps . . . 80
C Appendix - SU(2) identities 83
D Appendix - Critical points of n-invariants 85
E Appendix - EPRL Hessian 89
The program of Loop Quantum Gravity is to find a background independent quantum theory of general relativity. The goal is to be able in this way to give a proper formulation to problems that are not well posed in the context of perturbative quantum field theory, and ultimately to solve them. Examples are the fate of black holes and the resolution of the big bang singularity (and its effects on the cosmological power spectrum), which have attracted most of the efforts. The hypothesis that drives this search is that the failure of perturbative quantization to be effective at energy scales greater than the Planck mass does not need unknown extra degrees of freedom to be overcome; it would rather be due to the fact that quantum field theory, as we know it, relies strongly on Poincar´e symmetry: this symmetry cannot be preserved in a full quantum gravity setting, in which the causal structure should be dynamical. Expanding the gravitational field around a fixed flat spacetime collides with aiming to develop a theory which explains extreme gravitational phenomena. The strategy is to look for alternative quantizations of general relativity that do not spoil the geometric aspects of the classical theory, on the basis of the program of Dirac and Bergmann for the quantization of classical theories with constraints. In this perspective, in Loop Quantum Gravity the fundamental quanta are not gravitons, but quanta of space itself, whose geometry turns out to be non-commuting. They show discrete spectra, with minimal excitations proportional to the Planck scale. Our attention has been focused on the spinfoam approach, which provides a background independent path integral formulation for quantum gravity. The path integral provides the matrix elements of the projector on the kernel of the quantized Hamiltonian constraint and it is based on an action principle for general relativity due to Plebanski, which relates Einstein’s theory to a topological theory with internal gauge group SL(2, C) plus constraints. This local Lorentz group captures mathematically the equivalence principle. From a mathematical perspective, spinfoam amplitudes resemble a lattice gauge theory model, with the crucial difference that there is no background lattice spacing in the action. Amplitudes are expressed as integrals of group holonomies, and the constraints are implemented as a selection of the representation labels over which to sum. However, the more complicated dynamics and the absence of a lattice spacing make calculations much harder, and we still do not know whether the quantization has the correct continuum limit reproducing general relativity at low energies. The most interesting proposal up to now is the EPRL model (by Engle, Pereira, Rovelli and Livine), that provides well-defined transition amplitudes to all states in the Hilbert space of the theory. Works by Barrett and collaborators have shown that, performing a stationary phase approximation, the EPRL model has a clear semiclassical connection with a discretization of general relativity, Regge calculus, at least for the vertex amplitude.
In this thesis we review the spinfoam approach and its motivations, and present original results obtained in collaboration by Pietro Don`a, Simone Speziale, Giorgio Sarno and the author. We had a two-fold goal: on the one hand, to study the feasibility of a numerical evaluation of vertex amplitudes; on the other hand, to provide the first confirmation of the stationary phase approxi-mation of the vertex, and test its validity. This has led us to develop a Mathematica/C code for the numerical part and to obtain original results concerning the evaluation of SU (2) and SL(2, C) invariants, expanding the existing analytical techniques.
Chapter 1 reviews background and motivations for the Loop Quantum Gravity approach. We start from Einstein-Hilbert action and stress the importance of perturbative quantization of general relativity as an effective field theory, and its shortcomings. It would be important to know if Loop Quantum Gravity approaches reproduce the results of perturbative quantization, but for the moment the formalism does not permit easy comparisons. It is possible that perturbative and background independent quantization are apt to calculations in regimes that do not overlap. The Einstein-Cartan tetrad formalism is reviewed. The local Lorentz invariance and its breaking to a SU (2) subgroup are at the basis of loop quantization, as we see in the section dedicated to the construction of the Hilbert space of the theory as a representation of the holonomy-flux algebra. The role of the compact group SU (2) in a non-perturbative description of gravity is what leads to the discreteness of the spectrum of geometric observables. We also briefly comment on how to extract probability amplitudes in a general covariant setting. BF theory and Plebanski action of general relativity are presented: the simplicity constraints are shown to be the fundamental ingre-dient to obtain dynamics from the topological theory and the link with the generators of the local Lorentz transformation is established. Regge calculus is presented as the relevant discretization of general relativity to expect as a classical limit. Spinfoams are introduced as a sum over histories approach, and their scope in terms of computation of physical quantities is discussed.
Chapter 2 contains the definition a spinfoam model starting from a discretization of a BF theory and implementing a path integral quantization that uses delta function on group elements. We explain how to construct a foam and a partition function from a simplicial triangulation of a manifold. The SU (2) spinfoam theories are paradigmatic and we present them to explain the structure of spinfoam partition functions, with the help of a graphical notation that is used in the rest of the work and adapted to the SL(2, C) theory. SU (2) coherent states and their geometrical interpretation as semiclassical states are introduced.
Chapter 3 is a study of the vertex of the 4-dimensional SU (2) BF theory, the{15j}. In the first part we review the asymptotic analysis of the{15j} symbol, which reveals that for Euclidean boundaries the amplitude oscillates with the Regge action. We choose a path somewhat different in spirit from Barrett’s, concentrating on how to obtain the solution of the critical point equations starting from given boundary data. The expression of the action at all orders around the critical point and the calculation of the hessian are original contributions. The second section is about the numerical evaluation and it is completely original. It shows the first complete numerical computation of a 4d spinfoam amplitude, obtained solving the integrals in terms of Clebsch-Gordan coefficients, trading the problem of doing a multidimensional highly oscillatory integration with doing a sum of a very large numbers of terms, each of them easily computable. A careful choice of the intertwiner basis is crucial to make the computation feasible. We compare the data with the asymptotic formula and find that the asymptotic is reached quite fast for two different configurations we have considered.
Chapter 4reviews unitary irreducible representations of the SL(2, C) in the principal series, which are the main ingredient of the EPRL model. We present a canonical basis and the action of the generators on it, the construction of the Hilbert space on the linear space of homogeneous functions, and the coherent states to consider in this context. We explain how integrals of matrix elements can be written in terms of Clebsch-Gordan coefficients of SL(2, C), as we have these integrals in the EPRL model. As an original contribution, we perform an asymptotic analysis of a certain kind of invariant SL(2, C) symbols, which appear in the EPRL factorized vertex.
Chapter 5 presents the EPRL vertex. We explain how to extract the Y map, a selection on the representation labels, from the linear simplicity constraint. The vertex amplitude is first expressed as an integral on SL(2, C) of representation matrix elements, and then solved in terms of invariant symbols, and we extend the graphical notation to SL(2, C) objects. The expression of the integrals as a sum of Clebsch-Gordan coefficients is this time a convergent series, due to the
infinite dimension of Lorentz unitary irreducible representations. The factorization of the vertex amplitude shows that it can be written with a core given by the SU (2) amplitude and SL(2, C) invariant tensors at the boundary.
Chapter 6gets back to the initial question: are we ready to compute quantum gravity am-plitudes? If yes, can we recover the semiclassical limit? To answer these questions we first review the asymptotic analysis of the EPRL vertex, which reveals that for both Euclidean and Lorentzian boundary data the vertex admit critical points and oscillates, the difference between the two cases being the presence of the Immirzi parameter in the frequency, for the Lorentzian case. As for the SU (2) case, we concentrate on finding the critical points given a 4-simplex. The computation of the Hessian to which the chapter makes reference is an original contribution and it is given in AppendixE. Using the results on the asymptotic of SL(2, C) invariants, we individuate the terms in the series that are not exponentially suppressed in the large spin limit. Considering only these terms is a strong selection rule that may be effective for large spins, but testing this hypothesis numerically lies beyond our current computational power. We have searched evidence for the con-vergence of the amplitude at low spins and found confirmation, therefore we are confident to have effectively computed EPRL amplitude at low spins, both for Euclidean and Lorentzian boundary data. The comparison with the asymptotic formula is considered, and we conclude that only for Euclidean boundary data we are able to see the asymptotic behavior, even if not clearly as for the SU (2) amplitude. For Lorentzian boundary data the first points are dominated by the first term of the series, which is exponentially suppressed but still large at low spins, and it is impossible to find the asymptotic behavior. We finally comment on the significance of these results for realistic quantum gravity observables.
A path to quantum gravity
We present motivations for the loop quantum gravity approach illustrating the difficulties of per-turbative quantization and the program of canonical quantization. Tetrad formalism and general relativity in Ashtekar variables are reviewed, and we explain the construction of the Hilbert space of the theory as a representation of the holonomy-flux algebra. We comment on the discreteness of the spectrum of geometric observables and on the definition of probability amplitudes in a general covariant setting. We state the equivalence between Plebanski action and general relativity. Regge calculus is presented as the relevant discretization of general relativity to expect as a classical limit. Spinfoams are introduced as a sum over histories approach, and their scope in terms of computation of physical quantities is discussed.
1.1
Einstein-Hilbert action and perturbative
quantiza-tion
The best classical theory we have to describe gravitational phenomena is General Relativity, a field theory for a gravitational metric gµν, described in the pure gravity setting, without matter and
cosmological constant, by the Einstein-Hilbert action
SEH(gµν) = 1 16πG Z d4x√ −g gµνR µν(Γ (g)) . (1.1)
Here g is the determinant of the metric, and Γ(g) is the Levi-Civita connection defining the covariant derivative of vectors and tensors, the unique torsionless and metric compatible connec-tion. νµvν= ∂µvν+ Γνρµ(g)vρ, Γνρµ(g) = 1 2g νλ[∂ ρgλµ+ ∂µgλρ− ∂λgρµ]. (1.2)
When matter is coupled to gravitational field, the Einstein equations are Rµν−
1
2gµνR + gµνΛ = 8πG
c4 Tµν, (1.3)
where Λ is the cosmological constant and Tµν is the stress-energy tensor containing the matter
fields. Standard references are [Wald, 1984], [Weinberg, 1972], [Misner et al., 1973]
Even if up to now we have no experimental evidence of a quantum behavior of the gravitational field, it is reasonable to expect that there is a regime in which quantum effects are important,
even just because Tµν contains fields that we know possess quantum properties. Besides offering a
formal unification between the gravitational field and the other fields, a quantum theory of gravity is expected to solve unsettled dilemmas of the classical theory, such as the black hole and big bang singularities. This expectation is far more difficult to satisfy, while it is possible to write a quantum theory of gravity in the framework of effective field theories that is a perfectly acceptable quantum field theory. In fact quantum field theory offers a standard way to obtain a quantum theory from an action, that is the path-integral quantization. As a perturbative theory it organizes corrections to a quadratic field theory in powers of the coupling constant. The Einstein-Hilbert action is not polynomial in gµν, but one can write
gµν = ηµν+
1 Mpl
hµν, (1.4)
where ηµν is fixed to be the Minkowski metric, MP l2 = 16πG, consider hµν as the dynamical field,
and expanding in powers of MP l, obtaining an infinite sum of interactions. The quadratic part of
this lagrangian is the unique kinetic term for the embedding of a massless field of spin 2 in a field hµν, Lkin= 1 2hµνhµν− hµν∂µ∂αhµα+ h∂µ∂vhµν− 1 2hh, (1.5)
where h = hµµand indices are kept low for simplicity. Quite surprisingly, from this kinetic term one
can get back to Einstein-Hilbert action with relatively minimal requirements on the interactions with matter if asks for Lorenz invariance [Schwartz, 2014], [Feynman et al., 2003]. The crucial difference with the Standard Model fields is that the theory is non-renormalizable, as one can expect from the dimensions of the interactions, the leading interaction term having the form Lint=M1P lh3, and as it has been proved by Goroff and Sagnotti [Goroff and Sagnotti, 1986]. It
cannot be used to have predictions at energy scales near and above a certain dimensional parameter, in this case MP l. However, the unsettled questions about general relativity issues are expected
to happen just at those energy scales, therefore this theory of quantum gravity is insufficient. A possibility is that new degrees of freedom have to be introduced, as in the case of electroweak theory that extends Fermi theory, to UV complete quantum general relativity. String theory develops an approach of this kind, introducing also extra structures. Another possibility is that a different quantization approach to general relativity is preferable. Because of the strong reliance of QFT on Poincar´e symmetries, working around a flat space-time it may be too restricting, because in a non-perturbative context the causal structure is expected to be dynamical and show substantial quantum properties. Loop quantum gravity explores this possibility, with the hypothesis that avoiding the breaking of general covariance due to the expansion around flat space can lead to a theory capable of addressing non-perturbative issues.
1.2
Canonical quantization of constrained systems
The traditional approach to quantization of classical system in a non-relativistic setting is the canonical quantization. It can be applied also to field theories, but it is impractical because it spoils Lorentz covariance in the calculations. Its usefulness in the case of gravity lies in the fact that from a canonical point of view Einstein-Hilbert theory is a constrained theory where the Hamiltonian is a pure constraint, and Dirac and Bergmann [Dirac, 1964] established a program in the fifties to quantize constrained systems. Once one has established what are the canonical variables and the constraints from the classical theory, the quantization procedure follows in three steps
“kinematical” Hilbert spaceHkin, satisfying the standard commutations relations
{·, ·} −→ i~1 [·, ·] ; (1.6)
(ii) Promote the constraints to operators ˆHµ in
Hkin;
(iii) Characterize the space of solutions of the constraintsHphys,
ˆ
Hµψ = 0 ∀ψ ∈ Hphys. (1.7)
Canonical analysis of general relativity can be made in an ADM (Arnowitt, Deser and Misner) setting, explained for example in [Misner et al., 1973]. In this case the canonical coordinates are the 3-metric qab(x) and its conjugate momentum, πcd(x). The Dirac program in these variables
runs into technical difficulties. One of them is the lacking of an hermitian product: if one considers functionals of the 3-metric
ψ[qab(x)] (1.8)
formally one would need a hermitian product Z
dq ψ[q] ψ′[q]≡ hψ | ψ′i . (1.9)
However, there is no Lebesgue measure on the space of metrics modulo diffeomorphisms that we can use to define dq. Without this, we can not even check that ˆqab(x) and ˆπab(x) are hermitian,
nor that ˆqab(x) has positive definite spectrum, as needed for a spacelike metric.
Even if one ignores this issues, the quantization of the hamiltonian constraint is still problematic. One has what is known as the Wheeler-DeWitt equation:
ˆ Hψ[qab] = −~ 2 2G abcd: 1 √ detˆg δ2 δqab(x)δqcd(x) :−pdetˆgR(ˆg) ψ[qab], (1.10)
To give a meaning to this expression one has to find a suitable ordering prescription and regularize the differential operator, that is made by a product of operators at coincident points. However, there is not a characterization of the solutions in any sense.
The situations gets much more better if one choose Ashtekar variables, starting from an Einstein-Cartan formulation of general relativity. Introducing an ausiliary gauge invariance makes the constraints simpler and the definition of a kinematical Hilbert space possible.
1.3
Tetrad formulation
The first order Einstein-Cartan formulation of General Relativity in terms of tetrad and connection variables instead of the metric only is the starting point for both the modern canonical and covariant approach, and it let to overcome the difficulties presented in the last section. In this case the tetrad and the connection are varied independently. The action is polynomial in these variables, therefore constraints are simpler. A tetrad is a quadruple of 1-forms, eI
µ(x), I = 0, 1, 2, 3 such that
gµν(x) = eIµ(x)eJν(x)ηIJ. (1.11)
It defines locally a choice of coordinates which diagonalize the metric. In fact it defines one of a family of local inertial frames, related by a Lorentz transformation: the equation is fulfilled also for the transformed tetrad
In this way we have introduced an additional gauge symmetry, in the form of local Lorentz invariance. This collection of inertial frames at each point defines Lorentz principal bundle F = (M, SO(3, 1)) where the fiber over a point p is the set of the orthonormal bases of the tangent space at p related by a proper local Lorentz transformation. On this bundle we consider a connection ωIJ
µ , a 1-form with values in the Lorentz algebra, which defines a covariant derivative
that is compatible with the metric ηIJ
DµvI(x) = ∂µvI(x) + ωIµJ(x)vJ(x). (1.13)
This is the analogue of the covariant derivative νµ= ∂µ+ Γµfor vector fields in T (M). For objects
with both indices like the tetrad we write
DµeIν= ∂µeIν+ ωµJI eJν − ΓρνµeIρ. (1.14) The curvature is FIJ = dωIJ+ ωI K∧ ωKJ, (1.15) thus in components FµνIJ = ∂µωIJν − ∂νωIJµ + ωIKµωKJν − ωJKµωKIν . (1.16)
If ω is the Levi-Civita connection ω(e) the Einstein-Hilbert then
FµνIJ(ω (e))≡ eIρeJσRµνρσ(e), (1.17)
where Rµνρσ(e) is the Riemann tensor constructed out of (the metric defined by) the tetrad eIµ;
thus the action can be written as (recall we take units 16πG = 1), SEH(eIµ) = 1 2εIJKL Z eI∧ eJ ∧ FKL(ω(e)) . (1.18)
If we drop the ω(e) dependance we have
SEH(eIµ, ωµIJ) = 1 2εIJKL Z eI ∧ eJ ∧ FKL(ω) . (1.19)
The equation of motion in absence of matter impose the torsionless condition on ω, and given the uniqueness of the Levi-Civita connection, the equation of motion are solved by ω = ω(e). In the end they are equivalent to the Einstein equations
ωµIJ= eIν∇µeJν, Gµν(e) = 0. (1.20)
As it gives the same field equations, the theory defined by (1.19) is equivalent to general relativity up to some subtleties. One is the fact that (1.19) is defined for degenerate tetrads, while GR requires nondegenerate metrics because the inverse metric appear in the action. The other is that we can add another term with the right symmetries and mass dimension 4 - therefore giving in principle contribution at the leading order in perturbation theory - and it is:
δIJKLeI∧ eJ∧ FKL(ω), (1.21)
where δIJKL≡ δI[KδL]J. This term it is irrelevant in the ordinary second order metric formalism,
since when (1.17) holds,
δIJKLeI∧ eJ∧ FKL(ω(e)) = ǫµνρσRµνρσ(e)≡ 0. (1.22)
Adding this second term to (1.19) with a coupling constant 1/γ leads to the so-called Holst action [Holst, 1996] S (e, ω) = 1 2εIJKL+ 1 γδIJKL Z eI∧ eJ ∧ FKL(ω) . (1.23)
In addition to be zero on the solutions of the equations of motion, the term does not add any equation in the case of non-degenerate tetrads and absence of matter. This result is completely independent of the value of γ, that is an irrelevant parameter in the classical theory. However γ, that we call the Immirzi parameter, is crucial to write the action in terms of the Ashtekar variables and perform the canonical analysis that leads to quantization, as well as it establishes the linear simplicity constraint that is taken at the foundation of the covariant formulation. 1
Canonical variables can be extracted from1.24choosing a 1+3 foliation of the manifold M ≃ T× Σ. The action can also be written as
S (e, ω) = Z ∗ eI ∧ eJ +1 γe I ∧ eJ ∧ FIJ(ω)≡ Z ΠIJ∧ FIJ. (1.24)
One sees that the variables that appear with a time derivative are ωaIJ(~x), where a is only
a spatial index, and their conjugate momenta are ΠaIJ(~x), built from the pull-back of ΠIJ to
Σ.
The internal gauge symmetry is generated by a Gauss constraint. The generator of the local Lorentz transformations on a spacelike surface is
GIJ = ˆD
aΠaIJ = ∂aΠaIJ+ ωaΠaIJ+ ΠaωaIJ. (1.25)
With the aid of the normal nI to Σ, ΠIJcan be decomposed into dual and self-dual part
KI= nJΠIJ LI = nJ(∗Π)IJ. (1.26)
Since ΠIJ is antisymmetric n
IKI = nIJI = 0 and JI and KI are three dimensional vectors.
In the time gauge nI = (1, 0, 0, 0) their components are
Ki= Πi0 Li= 1 2ε
i
jkΠjk. (1.27)
From the fact that nIeI = 0, we also obtain
nIΠIJ= nI⋆ (eI∧ eJ) = γnI 1 γ⋆ (e I ∧ eJ) = γnI(⋆Π)IJ, (1.28) therefore ~ K = γ~L (1.29)
This property comes as a constraint from the Plebanski action that we will encounter. It is fundamental for the EPRL model and it is called the linear simplicity constraint. Since 1.25
generates local Lorentz transformations the fluxes of ~K and ~L inherit this role in the discrete theory and the linear simplicity constraint can be interpreted as a weak relation on the action of the generators in a representation of SL(2, C).
1
The Immirzi parameter becomes relevant also at the classical level if source of torsion are present [Perez and Rovelli, 2006,Freidel et al., 2005].
1.4
Quantization in new variables
Canonical loop quantum gravity is a quantization of general relativity using Ashtekar variables [Ashtekar and Lewandowski, 2004,Thiemann, 2008]. It shares with spinfoam models the kinemat-ical Hilbert space, and therefore it is useful to review its basics. In order to perform a canonkinemat-ical analysis one requires a 1+3 splitting of the spacetime manifold. Given a spacelike foliation Xtand
adapted ADM coordinates (t, x), we can define the time flow vector field τµ(x)≡∂X
µ t(x)
∂t = (1, 0, 0, 0), (1.30)
that we can decompose as
τµ(x) = N (x)nµ(x) + Nµ(x). (1.31)
where nµis the timelike unit vector vector to Σ and we have introduced the lapse function N and the
shift vector Nµ. It is convenient to parametrize nµ= (1/N,−Na/N ), so that Nµ= (0, Na).
In tetrad variables this translates into
eI0= eIµτµ= N nI+ NaeIa, δijeiae j
b = gab, i = 1, 2, 3. (1.32)
We call ”triad” ei
a, the spatial part of the tetrad. With respect to the general tetrad formulation,
this decomposition breaks local Lorentz covariance, but keeps rotational covariance in the internal indices. In particular, starting from Holst action, Ashtekar variables are a choice of canonical coordinates, built from the SL(2, C) connection and the tetrad, which transform as SU (2) objects, which is totally convenient from the point of view of the quantization. To keep notation simpler, we work in the ”time gauge” eI
µnµ = δI0, where
e0
µ= (N, 0)−→ eI0= N, Naeia . (1.33)
The Ashtekar-Barbero connection [Ashtekar, 1986,Barbero G., 1996] is Aia = γω0ia +
1 2ε
i
jkωjka . (1.34)
and its conjugate momentum is the densitized triad Eia = eeai =
1 2εijkε
abcej
bekc, (1.35)
In fact, we can rewrite the action (1.24) in terms of the new variables as [Barros e Sa, 2001,
Thiemann, 2008] S(A, E, N, Na) = 1 γ Z dt Z Σ d3xh ˙Ai aEia− Ai0DaEia− NH − NaHa i , (1.36) where Gj≡ DaEia = ∂aEja+ εjkℓAjaEaℓ, (1.37) Ha = 1 γF j abE b j− 1 + γ2 γ K i aGi, (1.38) H =hFabj − γ2+ 1 εjmnKamKbn iεjkℓEa kEℓb detE + 1 + γ2 γ G i∂ a Ea i detE. (1.39)
H(A, E) and Ha(A, E) are the Hamiltonian and space-diffeomorphism constraints, which are also
present in the traditional canonical analysis with the metric variables. The additional Gauss Constraint Gi comes from the additional gauge symmetry introduced by tetrad variables, and it
generates gauge transformations: Ejb and Aia transform respectively as an SU(2) vector and as an
SU(2) connection under the flow generated by the smearing of Gi
G (Λ) = Z
d3xGi(x)Λi(x). (1.40)
In fact its Poisson bracket with the canonical variables give Z d3xΛj(x)Gj(x), Eia(y) = Z d3xΛj(x)∂bEjb+ εmjnAmb Ebn, Eia(y)
= γεmjnΛj(y)Ebn(y)δbaδmi = γεijnΛj(y)Ean(y),
Z d3xΛj(x)G j(x), Aia(y) = Z d3xΛj(x)∂ bEjb+ εmjnAmb Ebn, Aia(y)
= γ∂aΛi(y) + γεmjiΛj(y)Ama (y).
The quantization program aims to construct a Hilbert space H in which the canonical com-mutation relations are realized, and find in it the subspace Hphys of the solutions Ψ to the
con-straints. ˆ GaΨ = 0 ˆ HaΨ = 0 ˆ HΨ = 0, (1.41)
The straightforward choice would be to pick H as the space of wave functionals of the con-nection, but there is not a way to build a hermitian product in this space that does not use any background dependent structure. To crucial step is to consider cylindrical functions, that are functions of the connection only trough a finite number of parallel transports, l ∈ Σ, called holonomies.
Ψf,Γ(A) = f (hl1(A) . . . hln(A)), (1.42)
Holonomies are SU (2) matrices, therefore a hermitian product is that given by the Haar measure of SU (2), and can be defined in a completely background independent way. This space can be decomposed in subspaces, each one of them is labelled by a graph Γ embedded in Σ,H = ⊗ΓHΓ.
For states on the same graph, the hermitian product is ψ(Γ,f )| ψ(Γ,f′) ≡
Z Y
e
dhef (he1[A], . . . , heL[A])f′(he1[A], . . . , heL[A]). (1.43)
This is easily extended to states in different graphs choosing another graph that contains both and integrating on that graph. The measure for this hermitian product is called the Ashtekar-Lewandoski measure
HΓ is a tensor product of L2(G, dµHaar). The Peter-Weyl theorem states that for a compact
group G the matrix elements of the unitary irreducible representations are a basis of the Hilbert space of functions on the group. In our case G = SU (2) and the decomposition
f (g) =X j ˆ fmnj Dmn(j)(g) j = 0,12, 1, . . . m =−j, . . . , j (1.44)
holds, where the Wigner matrix D(j)(g) is the representation of the group element g in the spin-j irreducible representation. A basis forHΓis obtained tensoring the basis vectors for each holonomy
space
hA | Γ; je, me, nei ≡ D(jm11)n1(he1) . . . D
(jn)
mnnn(hen), (1.45)
and a function ψ(Γ,f )[A]∈ HΓ can be decomposed as
ψ(Γ,f )[A] = X je,me,ne ˆ fj1,...,jn m1,...,mn,n1,...,nnD (j1) m1n1(he1[A]) . . . D (jn) mnnn(hen[A]). (1.46)
The restriction to cylindrical functions amounts to the quantization of a discretized version of the classical theory, obtained trough smearing the fundamental variables with test function. The smearing of the densitized triad corresponds to the flux operator on a surface S∈ Σ
E(S, α) = Z
S
αiE
i, (1.47)
The operators ˆhγ and ˆEi give the holonomy-flux algebra, and they are implemented in HΓ
following the classical Poisson parenthesis. In fact the holonomy-flux algebra is what survives, in the discretization, of the algebra of the canonical variables. If we had insisted with non-smeared operators we could not have defined the representation of the canonical algebra on the space of the cylindrical function, and we would be stuck with the necessity to use a choice of coordinates of the manifold, which makes background independence harder to keep.
The constraints need to be suitably smeared in order to be built from the holonomy-flux opera-tors, and this is not a trivial issue. For the Gauss constraint, we observe that it is the infinitesimal generator of gauge transformations at the classical level. The holonomy transforms with the gauge transformation at its endpoint, therefore inHΓwhat matters is the effect of a gauge transformation
at one node Λn, and we require that the action of Λn on a cylindrical function is the same function
evaluated on the transformed holonomies. The generator of this transformation is a multiple of the sum of the flux operators at the node n
ˆ Gin= 1 γ v X r=1 ˆ Eir (1.48)
where v is the number of link going into the node and Eir is the flux operator on the link r. This
can be seen as a smeared version of the Gauss constraint, where one considers only the leading term in the connection and and the densitized triad and transform the volume integral to a surface integral using Stokes’ theorem.
The states in the kernel of1.48are the spin network states. We indicate this Hilbert space as Hkin; it is the direct sum ofHkin,Γ, and each of these is isomorphic to L2(SU (2)L/SU (2)N, dµAL),
where L are the link and N the nodes of the graph. A basis of the gauge invariant space is given by intertwiners. The projector on the gauge invariant space at one node is given by the group averaging P = Z dgY e∈n D(je)(g). (1.49)
Since the integrand is an element in the tensor product of SU (2) irreducible representa-tions, Y e D(je) mene(he)∈ O e V(je). (1.50)
in general it belongs to representation that is the direct sum of several copies of irreducible repre-sentation. O e V(je)=M i V(ji). (1.51)
Only the projection on the copies of the trivial representation survives after the integration in (1.49), and we call this invariant space V(0). A basis of it is an intertwiner basis.
For the case of a 3-valent node, dimV(0)= 1 if and only if the triangular inequalities hold
|j2− j3| ≤ j1≤ j2+ j3. (1.52)
otherwise dimV(0)= 0. The unique intertwiner i corresponds in the three node space to a vector
that has a Wigner’s 3j-m symbols (cf. (1.54)) as components on the magnetic basis.
Pm1m2m3α1α2α3 = Z dg1Dm(j11)α1(g1)D (j2) m2α2(g1)D (j3) m3α3(g1) = j1 j2 j3 m1 m2 m3 j1 j2 j3 α1 α2 α3 . (1.53) For an n-valent node, the space V(0) can have a larger dimension. The case n = 4 is relevant for the simplicial graphs that are considered in the 4-dimensional covariant theory. In this case the intertwiners are given by the{4jm} symbols
Pm1m2m3m4α1α2α3α4 = Z dg1Dm(j11)α1(g1)D (j2) m2α2(g1)D (j3) m3α3(g1)D (j4) m4α4(g1) = X i di j1 j2 j3 j4 m1 m2 m3 m4 (i) j1 j2 j3 j4 α1 α2 α3 α4 (i) . (1.54)
Where i runs on every spin that satisfy the triangular inequalities with both the couples j1, j2
and j3, j4.
The covariant theory gives a prescription for the matrix elements on Hkin of the projector
on the kernel of the hamiltonian constraint, therefore we presented all the structure that we need. The diffeomorphism constraint is implemented as an action on the graph, reducing to the space of spin network modulo the equivalence relation for which two graphs are equivalent if they can be deformed into each other by the action of a diffeomorphism. The implementation of the hamiltonian constraint is the very hard part of the canonical approach, and there is not a conclusive solution. The discretization is not unique, and for the proposal that have been done there is not a characterization of the space of the solutions. We will not comment further on this issue because what follows is based on the covariant approach, in which there are other types of difficulties.
1.5
Observables and transition amplitudes
Observables are defined onHkinas self-adjoint operators and their eigenvalues can be used to label
states inHkin. Of fundamental importance is the Area Operator of a surface S∈ Σ
AS(E) =
Z
S
dx2Tr(nanbEaEb), (1.55)
where n is the co-normal. This operator inherits its geometrical meaning directly from the classical densitized triad. It is well defined [Rovelli and Smolin, 1995, Ashtekar and Lewandowski, 1997]
and it is diagonal in the spin-network basis, giving a clear interpretation to spin-network states. In fact its spectrum is given by
as({j}) = 8πγl2p
X
i
pji(ji+ 1) (1.56)
Therefore the area is In this picture area is quantized, bounded by below, and the quanta are proportional to the Planck length by the spins. One can also build a Volume Operator and it also has discrete spectrum. The volume operator ˆV (E) of a certain region in Σ has non zero eigenvalues for nodes with a valency n > 3. If area is concentrated on links, volume is on the nodes.
This, here briefly reviewed, is one the key results of LQG: the spacetime geometry is described by quantum operators with discrete spectrum, with eigenvalues proportional to the Planck length at the opportune power. It is also non-commutative: not all the intrinsic geometric observable commute and extrinsic curvature information is extracted by the holonomy operators, which do not commute with the fluxes.
Giving operational meaning to eigenvalues of operators (that is, values that can be measured) is a necessary step to give a probability interpretation to transition amplitudes. For a theory whose dynamics is given by a pure constraint, transition amplitudes are given by matrix elements of the projector ˆP on the kernel of constraint hψ| ˆP ψ′i. ψ and ψ′ are individuated by a collection
of values of observables and a probability amplitude can be extracted by the transition amplitude fixing some of these values. A transition amplitude can also be seen as the evaluation of a bilinear operator
W :H∗
f inal⊗ Hinitial→ C W (ψ⊗ ψ′) =hψ| ˆP ψ′i (1.57)
This definition generalizes to arbitrary boundary spacesH, not necessarily two different time slices. If Ψ(x, y)∈ H is a state labelled by observable values x and y the probability of measuring observable value ¯x after observing y is
P (¯x|y) = |W (Ψ(¯x, y)|
2
P
x|W (Ψ(x, y)|2
(1.58)
where the sum runs on all possible x. The expectation value of an operator O is obtained accordingly hOi|y = P xOx|W (Ψ(x, y))|2 P x|W (Ψ(x, y))|2 (1.59)
where Ox are eigenvalues of O and Ψ(x, y) the corresponding eigenvectors at fixed y. An
extended discussion about the problems of computing and predicting observables in a background independent theory is contained in [Rovelli, 2007]. 2
2
Note that this does not mean that translating to the ordinary prescription in quantum mechanics we can consider only commuting observables: in this context one has H = H∗
f inal⊗ Hinitialand considering Ψ(x, y) = |x, ti ⊗ |y, t′i
with t another observable with the role of time, P (x|y, t, t′) is the ordinary probability amplitude for finding x at
time t after given the measure of y at time t′, independently of the fact that the corresponding operators ˆXand ˆY
1.6
General relativity from BF theory
1.6.1
BF Theory: symmetries and classical equations
General Relativity in the first order formalism can be derived from a constrained BF theory. The constraints allow the existence of local degrees of freedom. The discretization of a BF Theory is the starting point for the spinfoam quantization, which has a straightforward and unambiguous definition for a topological theory. The quantization procedure for a gravity theory has more subtleties involving the implementation of the linear simplicity constraint, and it is described in the following chapter. Here we review some aspects of the classical theory.
First, we recall the definition of a BF Theory following the approach in [Baez, 2000] and [Perez, 2013]. We consider the case of a compact group G whose a Lie algebra g has an in-variant inner product here denotedh i, and M a d-dimensional manifold. Classical BF theory for the group G is defined by the action
S[B, ω] = Z
M
hB ∧ F(ω)i, (1.60)
where B is a g valued (d− 2)-form, ω is a connection on a G principal bundle over M. If G is semisimple the bilinear form can be taken to be the Killing form, that is hx, yi = Tr(xy), where the trace is taken in the adjoint representation. The action displays is a manifest invariance under local G gauge transformations
δB = [B, α] , δω = dωα, (1.61)
where α is a g-valued 0-form.
In fact, setting the variation of the action to zero 0 = δS = Z M Tr(δB∧ F + B ∧ δF ) = Z M Tr(δB∧ F + B ∧ dωδω) = Z M Tr(δB∧ F + (−1)d−1dωB∧ δω), (1.62)
where dω stands for the exterior covariant derivative, that acts on a (1,1) tensor valued p-form f
as
dωf = df + ω∧ f + (−1)p+1f∧ ω
We have also used that dωTr[B∧ δω] = d Tr[B ∧ δω] in order to use Stokes’ theorem and drop
boundary terms.
We see that the variation of the action vanishes for all δB and δω if and only if the following field equations hold
F = 0, dωB = 0. (1.63)
A less evident symmetry is given by the transformations
for some ad(P)-valued (n − 3)-form η. Indeed the action is left invariant up to boundary terms: Z M Tr((B + dωη)∧ F ) = Z M Tr(B∧ F + dωη∧ F ) = Z M Tr(B∧ F + (−1)dη∧ dωF ) = Z M Tr(B∧ F ), (1.65)
These symmetries imply that all the solutions are equivalent up to a gauge transformation, and in particular they are locally equivalent to the trivial ω = 0, B = 0 solution. In fact F = 0 says the connection ω is flat and all flat connections are locally the same up to gauge transformations. Locally we can choose ω = 0, and then dωB = dB = 0, therefore B = dωη for some η because
locally all closed forms are exact. Using the B 7→ B − dωη we obtain B = 0. We observe that
there are no local degrees of freedom and the characterization of a solution can be only global, or topological. This is what we mean when we say that the theory is topological. If we specialize G = SU (2) and d = 3 we recover Riemannian General Relativity in which the field Bi
a is given by
the tetrad frames ei a.
1.6.2
Plebanski action
From a BF theory G = Spin(4) d = 4 we can recover Riemannian General Relativity in d = 4, in the first order formulation, adding to the action a constraint. For simplicity of notation we will treat the Riemannian case only; the Lorentzian case is completely analogous, the only difference is in the correct signs in the contractions of the indices. Consider a Lie-algebra-valued 2-form B, an so(4) connection ω and a Lagrange multiplier λ. The Plebanski action is given by
S[B, ω, λ] = 1 κ Z M (⋆BIJ+1 γB IJ) ∧ FIJ(ω) + λIJKLBIJ∧ BKL , (1.66)
where we have F , the curvature of the connection ω, a constant κ and λIJKL =−λJIKL =
−λIJLK = λKLIJ that is a tensor in the internal space satisfying ǫIJKLλIJKL = 0. γ is a free
parameter, called the Immirzi parameter; its role is explained in the next paragraph. The hodge is given by∗BIJ =1
2ǫ IJ
KLBKL. The equation of motion for this action add further conditions on
the variables with respect to pure BF , coming from the variation of the Lagrange parameter λ. On the 36 components of the field B, we have 20 algebraic equations that are
ǫµνρσBµνIJBρσKL= e ǫIJKL, (1.67)
where e = 4!1ǫOP QRBµνOPBρσQRǫµνρσ [De Pietri and Freidel, 1999]. These equations are solved if B
has the form
BIJ=±∗(eI
∧ eJ), and BIJ=
±eI
∧ eJ, (1.68)
that is it can be expressed in terms of the remaining 16 degrees of freedom of a tetrad field eI a. If
we use the first solution in the action1.66we obtain Holst action [Holst, 1996] S[e, ω] = 1 κ Z M Tr (⋆e∧ e +γ1e∧ e) ∧ F (ω) . (1.69)
As we have seen Holst action is the starting point for the canonical quantization approach. There-fore Plebanski action shows the link between the covariant approach of spinfoam quantization and the canonical one.
The canonical analysis of the Plebanski action is performed in [Perez, 2013] and it is shown to be equivalent to that in Ashtekar variables, once the constraint are solved. A subset of the constraints is equivalent to5.1and one of these constraint is the linear simplicity constraint.
1.7
Regge calculus
In this section we state some elements of Regge theory for discretized spacetimes, which is useful to understand the classical limit of spinfoams. In Regge theory the continuous spacetime, the d dimensional manifold, is replaced by a discretization made by d-simplices. For instance, in d = 2 we can triangulate our surface with triangles, in d = 3 with tetrahedra and in d = 4 with 4-simpleces. The geometry, and so the continuous metric, is recovered by the areas of the d− 1 elements of the triangulation and by the deficit angle that measure how far we are from a flat situation. In d = 2, for example, our triangulation is done with triangles glued together. A certain point P will be a vertex for a a certain number of triangles t. The deficit angle at this point is defined as
δP(ls) = 2π−
X
t
θt(ls) (1.70)
where the sum is performed over each triangle surrounding P ,θt is the angle at P of the triangle
t and ls are the length of the edges of the triangles. If the deficit angle is 2π it means that the
Regge space is flat in P . Using just areas and deficit angles we can define the action for the d dimensional Regge space (M, ls)
SR(ls) =
X
h=(d−1)simpleces
Ah(ls)δh(ls). (1.71)
The remarkable result is that the above equation leads to the Einstein-Hilbert action in the con-tinuum limit, when the Regge space converges to a Riemannian manifold. Regge theory is a good discretization of Einstein’s theory of gravitation. The hope to find a quantum theory from this discretization of general relativity dates back from the late ’60s, when Ponzano and Regge built a model [Ponzano and Regge, 1968] for 3d discrete gravity in which the manifold is triangulated by tetrahedra with spins jton each edge ls. They argued that{6j} symbol, that we will define later,
converges to the 32d Regge action in the semiclassical limit in which spins are large, and thus to 3d General Relativity in the continuum limit. This property, finally proved in 1998 by Roberts [Roberts, 1999], states that the leading order of the asymptotic expansion is
{6j} ∼ √ 1 12πV cos (SR+ π 4) = 1 2√−12iπVe iSR+ 1 2√12iπVe −iSR (1.72)
where V is the volume of the tetrahedron. The four dimensional analogue of this model has waited for almost half of a century 3 An important part of the this work is dedicated to review
how spinfoam models are the realization of the ideas of Regge and Ponzano in a 4 dimensional setting.
3
A coherent path integral approach to Regge Calculus has been developed [Roˇcek and Williams, 1984]. However it encounters problems of regularization and it has no notion of a well-defined Hilbert space. In this sense spinfoam theory could be seen as a quantum Regge Calculus regularized by the minimal area and with a precise notion of geometrical quantum states on an Hilbert space.
sΓ,{jl},{in}
s′ Γ′,{j′
l},{i′n}
Figure 1.1: An example of 2-complex J . The nodes of the spin network become edges of the 2-complexJ and they carry intertwiners while the faces generated by spins are labelled by irreducible representations jf.
1.8
Spinfoams as path integrals for generally covariant
sys-tems
The idea to define a path integral for the gravitational field has been considered since Misner [Misner, 1957] in the late ’50s. Consider a manifoldM with two boundaries Σ1and Σ2. Formally,
one would like to define the transition amplitude between|qabi on Σ1 and|q′abi on Σ2
hqab|q′abi =
Z
g|q′ ,q
D(g)eiS(g). (1.73)
However there is no well-defined notion ofD(g). The best developed approach following this idea, Hawking’s Euclidean quantum gravity, has so far had only limited applications to minisuperspace models.
The spinfoam formalism that we are going to present in the next chapter proposes a definition of this path integral as a sum over histories of quantum geometry, namely over spin network states. These spin network world sheet are called spinfoams. It is intended as a prescription for the matrix elements of the projector on the kernel of the constraints. The general structure of spinfoam formalism has a precise mathematical definition in the sense of category theory due to [Baez, 1998]. For our purpose we state that a spinfoam can be seen as an object that connects two spin networks ψΓ,{jl},{in}, ψΓ′′,{j′
l},{i′n}
F : ψ → ψ′ (1.74)
It consist on an assignment to a 2-complexJ with boundaries given by ψ and ψ′as in figure1.1of
representation labels jf for every f ∈ J and intertwiners in to the edges e∈ J , compatibly with
the boundary labels. IfA(F) is the evaluation if this assignments, a spinfoam amplitude is given by
hs|s′i = X
F:s→s′
A(F), (1.75)
where the sum is over all possible assignmentsF. It satisfies a composition rule
A(F ◦ F′) =A(F)A(F′). (1.76)
The 2-complex can be thought as the spacetime, where the spin networks are the space slices and the geometry is encoded in the labels attached toJ .
Given this general structure, different models have been studied in the literature. The situation can be loosely compared to a Lattice Gauge Theory, where different choices of plaquette action can be acceptable, provided a continuum limit exists. The fundamental difference is that in a general covariant theory the lattice spacing does not appear in the action, and the continuum limit is just the limit in which the number of the links and edges of the foam goes to infinity,
without having to tune a parameter. In this the theory is formulated as a perturbation theory in the number of degrees of freedom considered. One can postpone this foundational issues trying to extract predictions already from the most simple graphs and check with experiments, if they were possible. However, one must not confuse the continuum limit with the classical limit, which is the limit of large quantum numbers and it is also unsettled. Being spinfoam theories discretized models, the classical limit at fixed graph cannot be general relativity but at most a discretized version of it, like Regge calculus. In the following, we will focus on a model that is currently the most promising one, the EPRL model. The justification and the geometric interpretation of the model is based on GR as a constrained BF theory. One of the main reasons for which this model is promising is that for the most simple spinfoams the semiclassical limit is proven to be Regge calculus. An important comment is that ultraviolet divergences are absent, but there could be infrared divergences in foams with internal faces. In [Riello, 2014] some self-energy divergencies are analyzed. The regularization of these divergences is still an open problem. A solution may be, in the spirit of Turaev-Viro model [Turaev and Viro, 1992], to deform SL(2, C) to its quantum group version. This move can be justified as the introduction of the cosmological constant, which would be a natural regulator.
The advantage of spinfoam models with respect to canonical quantization is that in the latter one has to characterize the kernel of the constraint in order to find the projector. The downside is that spinfoam quantization of a classical system is less under control on formal grounds. This is of relative importance as soon as one can extract predictions from spinfoam models. Because of the precise prescription for amplitudes in spinfoam models with respect to the convolute procedure one should follow in canonical theory, the evaluation of transition amplitudes may be just tackled as a problem of computational complexity. N-point functions can be computed in the context of spinfoam models, as in [Bianchi and Ding, 2012], [Bianchi et al., 2009], but it is not clear how to compare them with the effective field theory calculations, because they stuck in a discretized setting. An example of observable that could be computed with spinfoam amplitudes is the time for the bounce of a black hole as seen from an external observer, using the amplitude for the flip of the extrinsic curvature of a 3d ball [Christodoulou et al., 2016].
In this work we explore the possibility of computing numerically the amplitude and search for the evidence of the classical behavior for large spins. We believe that this is an important step to understand if the theory can explain us something about quantum gravity.
Spinfoam quantization
We review the quantization of a discretized BF theory on a gauge group G and present a graphical notation for spinfoam amplitudes. We consider SU (2) models in three and four dimensions as paradigmatic examples. We introduce coherent states and their role in probing the semiclassical regime of spinfoams.
2.1
Spinfoam quantization of BF theory
We consider a BF theory on a compact group and an orientable manifold. With its action we can formally build a partition function from the path integral
Z = Z DBDω exp(i Z M BIJ∧ FIJ(ω)). (2.1)
We can integrate over the B field obtaining Z =
Z
Dω δ (F (ω)) . (2.2)
We need a regularization in order to give sense to this expression. In a sprit similar to lattice gauge theory, we can regularize the expression with a truncation, considering variables associated to discrete paths and integrating on them. This truncation is also necessary for the theory to be compatible with the Hilbert space of Loop Quantum Gravity.
In order to discretize our variables we replace the manifold M with an arbitrary cellular de-composition ∆. We introduce also the notion of dual 2-complex of ∆ denoted by ∆⋆ The dual
2-complex ∆⋆ is a combinatorial object defined by a set of vertices v
∈ ∆⋆ (dual to d-cells in ∆)
edges e ∈ ∆⋆ (dual to (d
−1)-cells in ∆) and faces f ∈ ∆⋆ (dual to (d
−2)-cells in ∆). In the proceeding we think of ∆ as a triangulation. The field B is discretized with Lie algebra elements Bf associated with the faces∈ ∆⋆, corresponding to the smearing of the continuous (d− 2) form
B on the (d− 2) cell in the triangulation ∆ (e.g. on the faces of the dual triangulation ∆⋆). More
precisely
Bf =
Z
(d−2)cell
B. (2.3)
In the same manner the connection ω is discretized with group elements ge ∈ G assigned to the
edges of ∆⋆. These group elements are the holonomy of ω along the edge e∈ ∆⋆.
ge= P exp(−
Z
e
ω), (2.4)
where P stands for the path-ordered product. With all this in mind, the discretized version of the path integral (2.1) is Z(∆) = Z Y e∈∆⋆ dge Y f ∈∆⋆ dBf eiBfUf = Z Y e∈∆⋆ dge Y f ∈∆⋆ δ(ge1· · · gen), (2.5)
where we have Uf = ge1· · · gen representing the holonomy around a face. This equations encodes
exactly the discretized version of (2.2). The integration done in Bf is performed using the group
delta function. The remaining integration in the holonomy variables is done in term of the invariant measure of G that, when G is compact, is a unique Haar measure. We can then expand the delta function using the Peter-Weyl theorem obtaining
δ(g) =X
ρ
dρTr[ρ(g)], (2.6)
where ρ are irreducible unitary representations of G and dρ the dimensions of the vector spaces
given by the representations ρf. From the previous expression one obtains
Z(∆) =X {ρ} Z Y e∈∆⋆ dge Y f ∈∆⋆ Tr[ρf(ge1. . . gNe )]. (2.7)
Now we can also perform the integration over the discretized connection using the fact that in a triangulation ∆ the edges e bound exactly d different faces. So, every group element geappears in
d traces and we can use the following formula that defines the projector over the invariant states of the theory
Pinve (ρ1, . . . , ρd) :=
Z
dgeρ1(ge)⊗ ρ2(ge)⊗ · · · ⊗ ρd(ge), (2.8)
where the representations matrices ρ have a precise indices contraction given by the structure of the triangulation. The tensor product of d ρ matrices define a space H = Hρ1 ⊗ . . . Hρd that has
an invariant part under the group G considered. Pe
inv is precisely the projector over these invariant
states living in Inv[Hρ1⊗ Hρ1⊗ . . . Hρd]. Finally we have, for a BF theory with a compact group,
the following partition function ZBF(∆) = X {ρ} Y f ∈∆⋆ dρf Y e∈∆⋆ Pinve (ρ1⊗ · · · ⊗ ρd). (2.9)
This is the simple discretized BF partition function, a sum over irreducible representations assigned to the faces of ∆⋆ of the number given by the contraction of the projectors in a way given by the
structure of the two-complex. There is a very beautiful and extremely useful graphical notation for this construction. We are going to use it extensively during the following because it allows a very easy way of building amplitudes and it makes thing much more clean and clear. This notation has been introduced by Oeckl [Oeckl and Pfeiffer, 2001, Oeckl, 2005] and developed by [Perez, 2013] and [Speziale, 2017]. We will use also many techniques for quantum angular momentum theory as in [Varshalovich, 1988]. In AppendixAthere is a glossary relating all graphical structures to their analytical expression.
The idea is to represent every representation in2.7 by a line (wire) labeled and every integration with a box. In this notation the projector becomes
Pinve (ρ1⊗ ρ2⊗ · · · ⊗ ρd) =
: : :
: : : ρ1ρ2 ρd
. (2.10)
The contraction of d irreducible representations projected on the subspace of invariant states.
Fix now d = 2. A triangulation is made by triangles glued together. At every vertex of a triangle we associate a representation ρi and at every edge of the same an integration and so a
box. In fact, every edge of a triangle has two vertex and so, if we go in the dual picture, every e∈ ∆⋆ is shared by two faces. This means that at the elements g
eare assigned two representations
(namely at one box are assigned two wires). The result of the BF amplitude is, in this case
ZBF(∆) = X {ρ} Y f ∈∆⋆ dρ ρ1 ρ2 ρ3 , (2.11)
the open ends on the right side are linked to neighboring vertices according to the structure of the triangulation. The object on the right side is the basic building block for our 2D triangulation. If we imagine this structure for a triangulation in d = 3, ∆ will be made by tetrahedra. In this case we need just to go up in dimension: a box is associated with a triangle and every line is associated with the segments of this triangle. This means that in the dual picture at every edge e (dual to a triangle) we have three different faces f (dual to the segments of the triangle). At every box three wires. As we did before we can write down the BF amplitude
ZBF(∆) = X {ρ} Y f ∈∆⋆ dρ ρ1 ρ2 ρ3 ρ4 ρ5 ρ6 , (2.12)
Finally in d = 4 the basic object of a triangulation is a 4-simplex made by five tetrahedra glued together by triangles. In the dual picture now every edge is touched by five faces and so
ZBF(∆) = X {ρ} Y f ∈∆⋆ dρ ρ1 ρ2 ρ3 ρ4 ρ5 ρ6 ρ7 ρ8 ρ9 ρ10 . (2.13) To summarize we have
Spacetime Dimension Triangulation ∆ Assignment
d=2 Triangles Edges7→ Boxes
Vertices7→Wires
d=3 Tetrahedra Triangles7→ Boxes
Edges7→Wires
d=4 4-Simplices Tetrahedra7→ Boxes
Triangles7→Wires
This completes the general setting for a BF theory in d dimensions with compact G symmetry group. The idea is to study this objects in the cases useful for gravity. We now look at two explicit
toy models that are useful to see how to implement this ideas and how to rewrite the amplitudes in a more convenient way specializing them to 3 and 4 dimensions.
2.1.1
Riemannian 3d Gravity: SU(2) BF Theory
Fix now d = 3 and G = SU (2). In three dimensions every edge of ∆⋆is shared by three faces. So the
projector Pe
inv takes the contraction of three representations, labeled by three spins jf projecting
them to the invariant part of the tensor product of the vector spaces of the three representation. As we have already seen it can be written explicitly in terms of Wigner 3jm symbols as
Z dg 3 O i=1 Dji mi,ni(g) = j1 j2 j3 m1m2m3 j1 j2 j3 n1n2n3 , (2.14)
where we are using the conventions reported in the appendix B. This equation graphically becomes = j1 j1 j1 j1 j2 j2 j2 j2 j3 j3 j3 j3 (2.15)
where the 3-valent open graphs on the right hand side represent the unique normalized invariant vector inH, namely the 3jm symbol. If we use this identity in2.17we get the following expression for Z ZBF(∆) = X {jf} Y f djf j1 j2 j3 j4 j5 j6 (2.16)
where open strands are linked to other tetrahedra. We can finally rewrite the amplitude as a product over vertices (tetrahedra) of our triangulation each one of them is given by a symbol as in the right side of the formula above then we will simply call{6j} symbol. We get
ZBF(∆) = X {jf} Y f djf Y v j1 j2 j3 j4 j5 j6 . (2.17)
The amplitude is given by a sum over all possible of irreducible representation of SU (2) at the faces of the 2-complex ∆⋆ weighed by the dimensions of the vector spaces of the representations
assigned to the face and then a product of {6j} symbols, one for each vertex of the 2-complex, that is of each block of the triangulation.
As we saw in the introduction the state sum presented above is related to 3d gravity via the Ponzano-Regge model of the 1960s [Ponzano and Regge, 1968,Barrett and Naish-Guzman, 2009]. The asymptotic expression of the {6j} symbol is related to the action of the discretization of General Relativity. One of the main goal of this thesis is exactly to numerical study the asymptotic of this object in 4d, first considered by [Ooguri, 1992].
2.1.2
4d SU(2) BF Theory
To construct the model in 4 dimensions we use the same ideas. This time every edge shares four representations, and the invariant space is spanned by 4jm symbols. We recall that the projector is given by Z dg 4 O i=1 Dji mi,ni(g) = X i dji j1 j2 j3 j4 m1m2m3m4 (i) j1 j2 j3 j4 n1n2n3n4 (i) (2.18) where the sum over the intertwiners i goes from Max(|j1 − j2|, |j3 − j4|)) to Min(j1 + j2, j3 + j4). Graphically =
P
i i j1 j1 j1 j1 j2 j2 j2 j2 j3 j3 j3 j3 j4 j4 j4 j4 di (2.19)Given this we can rewrite the partition function as
ZBF(∆) = X {jf},{ie} Y f djf Y e die j1 j2 j3 j4 j5 j6 j7 j8 j9 j10 i1 i2 i3 i4 i5 , (2.20)
where for each box we obtain a summation over the intertwiners and, as we did in the three dimensional case, we can rewrite the formula above as
ZBF(∆) = X {jf},{ie} Y f djf Y e die Y v j1 j2 j3 j4 j5 j6 j7 j8 j9 j10 i1 i2 i3 i4 i5 . (2.21)
The symbol on the right side is called{15j} symbol or vertex amplitude and it is the main building block that we will investigate in the proceedings. It is the main ingredient of the spinfoam model and it is the starting point to investigate the semiclassical limit of the theory as we do in chapter
3. We will show that a single{15j} evaluated on coherent states reproduces the Regge action for 4d Riemannian gravity in the semiclassical limit.
This concludes our presentation of compact models for a BF theory. Another interesting model that we will not review is the Riemannian model based on the group G = SU (2)×SU(2) ⋍ SO(4), from which one can construct a theory for Euclidean gravity. It is essentially constructed by gluing two {15j} symbols by some factors called fusion coefficients. Further details in [Perez, 2013].
2.2
Coherent States
Coherent states for spinfoams amplitudes have been investigated by Thiemann and col-laborators [Thiemann, 2001, Thiemann and Winkler, 2001a, Thiemann and Winkler, 2001b] and their importance for the spinfoam theory has been emphasised by Livine and Speziale in [Livine and Speziale, 2007] as coherent states of intertwiners. This ideas are related to the Barberi quantum tetrahedron [Barbieri, 1998].
We introduce coherent states in the case of SU (2) analysis starting from the representation space Hj of dimension dj ≡ 2j + 1. On this space we have a canonical orthonormal basis |j, mi
and we can write the resolution of the identity in term of this basis 1j=
X
m
|j, mihj, m| (2.22)
where −j ≤ m ≤ j. We can define states |j, gi ∈ H called SU(2) coherent states that are defined by the action of a group element on the maximum state |j, ji
|j, gi ≡ g|j, ji =X
m
|j, miDmjj (g) (2.23)
With this states we can rewrite2.22as X m |j, mihj, m| = dj X mm′ |j, mihj, m′| Z SU(2) dg Dmjj (g)Djm′j(g) = dj Z SU(2) dg|j, gihj, g| = 1j (2.24)
where at the first step we have used the orthonormality of the matrix elements. Noticing now that Djmj(g) and D
j
mj(gh) differ only by a phase for any group element h from a suitable U (1)⊂ SU(2)
we can express the relation on coherent states above in term of an integral on the two-sphere of directions S2= SU (2)/U (1). We obtain
1j= dj
Z
S2
dn|j, nihj, n| (2.25)
in which n∈ S2 is integrated with the invariant measure of the sphere. Our new states|j, ni form
an over-complete basis in Hj and they have the usual semiclassical properties. If we consider the
operators Ji, generators of su(2), we get
hj, n| ˆJi
|j, ni = j ni (2.26)
where ni is the corresponding three dimensional unit vector for n
∈ S2. The fluctuations of ˆJ2
are also minimal with ∆J2 = ~2j, where we have restored ~ just for clarity. In the limit ~
→ 0 and j → ∞ the fluctuations go to zero while ~j is kept constant. This is what we will call the semiclassical limit in spinfoams and with this state we will study the asymptotic of the model. So the state |j, ni is a semiclassical state describing a vector in R3 of length j and of direction n. As
we did in the previous cases we can introduce also for this states a graphical notation. Equation
2.25becomes j = dj Z S2 dn j n (2.27)
These states have another important property deriving from the fact that we can build a representa-tion ρjas the symmetric tensor product of 2j times the fundamental representation, namely
|j, ji ≡ |12,1 2i
and so
|j, ni = |12, ni⊗2j (2.29)
These states are fundamental in the study of the semiclassical limit of spinfoam theory and we can also parametrize them in the following way
|j, ni = Dj(h(n))|j, ji (2.30)
where D1(h(n))ˆz = n and we use the convention Dj(h(n)) = e−iJzφeiJyθeiJzφ, φ and θ are Euler
angles. A coherent tetrahedron state is determined by the boundary data{ja, na}, a ∈ {1, 2, 3, 4}
as |{ja, na}i = Z SU(2) dg 4 O a=1 Dja(gh(n a))|ja, nai (2.31)
We have now a coherent picture in which every spins is related to the area of one of the triangle forming the tetrahedron and each ~n is a vector in R3representing the normal at that face. These
are the coherent states we are going to use as boundary data. The partition function can be specialized to these boundaries as we do in the next section.
2.2.1
Amplitudes in the Coherent Representation
We want to rewrite the path integral for a BF Theory defined in section2.1.2in the coherent state basis. In order to do that we simply insert a resolution of the identity on every wire connecting two neighbouring vertices of the triangulation obtaining
ZBF(∆) = X {jf} Y f djf Z Y e djefdnef n1 n2 n3 n4 (2.32)
where we have written just the ni∈ S2 on a single box but they are meant to be on every one. In
this semiclassical picture one can think at the spins ji associated to the wires as the areas of the
triangles forming a tetrahedron and at the nias the geometrical 3d normals to these triangles. At
every box (tetrahedra) we have four representations (triangles) and this pictures is completed by the coherent states, at fixed box one for each wire. In this sense the sum over the spins represents a superposition of all possible combination of tetrahedra inside the foam and the integral in dn as a superposition of the normals. We also rewrite this expression analytically because it will play an important role in the next chapter. Rewriting the boxes as integrations we get the following analytical expression for the partition function
ZBF(∆) = X {jf} Y f djf Y v Z Y e djefdnef Z Y e,e′∈v dgevhJnef|gevg−1e′v|ne′fi2jf =X {jf} Y f djf Y v Z Y e djefdnef{15j}(jf, nf) (2.33)
where we have denoted gev the group element in the v-simplex assigned to the edge (box) e, we
have used the property 2.29and J denotes the standard anti-linear SU (2) operator. With this expression we can start our analysis for the amplitude associated with a single coherent 4-simplex with assigned boundary data.
